2025/09/22 by Kai Yuan, Yan Wang, Yuan, Kai +1
Computer Science · Mathematics · #05C25 #05C30 #05C62 #05E16 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2509.17910
openalex publication_date 2025/09/22 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
It is well-known that every vertex-transitive graph admits a representation as a coset graph. In this paper, we extend this construction by introducing monodromy graphs defined through double cosets. Our main result establishes that every graph is isomorphic to a monodromy graph, providing a new combinatorial framework for graph representation. Moreover, we show that every graph gives rise to an arc-transitive graph through its monodromy representation. Inspired by the monodromy representation of graphs, we denote an algebraic map M(G;Ω,ρ,τ) by M(G;U,ρ,τ) where U is a stabiliser in G. As an application, we prove an enumeration theorem for orientable maps with a given monodromy group. We underscore a fundamental triad in algebraic graph theory: Where there is a graph, there is a group, an arc-transitive graph, and an orientable regular map--each arising canonically from the underlying combinatorial and algebraic structures.